Solve a^-2+b^-2/a^-2-b^-2

$$\frac{ { a }^{ -2 } + { b }^{ -2 } }{ { a }^{ -2 } - { b }^{ -2 } }$$

Evaluate

$$\frac{a^{2}+b^{2}}{b^{2}-a^{2}}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{\left(b^{-2}a^{2}+1\right)a^{-2}}{\left(-b^{-2}a^{2}+1\right)a^{-2}}$$

Cancel out a^{-2} in both numerator and denominator.

$$\frac{b^{-2}a^{2}+1}{-b^{-2}a^{2}+1}$$

Expand the expression.

$$\frac{1+\left(\frac{1}{b}a\right)^{2}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a as a single fraction.

$$\frac{1+\left(\frac{a}{b}\right)^{2}}{1-\left(\frac{1}{b}a\right)^{2}}$$

To raise \frac{a}{b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1+\frac{a^{2}}{b^{2}}}{1-\left(\frac{1}{b}a\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

$$\frac{\frac{b^{2}}{b^{2}}+\frac{a^{2}}{b^{2}}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a as a single fraction.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{1-\left(\frac{a}{b}\right)^{2}}$$

To raise \frac{a}{b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{1-\frac{a^{2}}{b^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{\frac{b^{2}}{b^{2}}-\frac{a^{2}}{b^{2}}}$$

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{\frac{b^{2}-a^{2}}{b^{2}}}$$

Divide \frac{b^{2}+a^{2}}{b^{2}} by \frac{b^{2}-a^{2}}{b^{2}} by multiplying \frac{b^{2}+a^{2}}{b^{2}} by the reciprocal of \frac{b^{2}-a^{2}}{b^{2}}.

$$\frac{\left(b^{2}+a^{2}\right)b^{2}}{b^{2}\left(b^{2}-a^{2}\right)}$$

Cancel out b^{2} in both numerator and denominator.

$$\frac{a^{2}+b^{2}}{-a^{2}+b^{2}}$$

Expand

$$\frac{a^{2}+b^{2}}{b^{2}-a^{2}}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{\left(b^{-2}a^{2}+1\right)a^{-2}}{\left(-b^{-2}a^{2}+1\right)a^{-2}}$$

Cancel out a^{-2} in both numerator and denominator.

$$\frac{b^{-2}a^{2}+1}{-b^{-2}a^{2}+1}$$

Expand the expression.

$$\frac{1+\left(\frac{1}{b}a\right)^{2}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a as a single fraction.

$$\frac{1+\left(\frac{a}{b}\right)^{2}}{1-\left(\frac{1}{b}a\right)^{2}}$$

To raise \frac{a}{b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{1+\frac{a^{2}}{b^{2}}}{1-\left(\frac{1}{b}a\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

$$\frac{\frac{b^{2}}{b^{2}}+\frac{a^{2}}{b^{2}}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{1-\left(\frac{1}{b}a\right)^{2}}$$

Express \frac{1}{b}a as a single fraction.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{1-\left(\frac{a}{b}\right)^{2}}$$

To raise \frac{a}{b} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{1-\frac{a^{2}}{b^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b^{2}}{b^{2}}.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{\frac{b^{2}}{b^{2}}-\frac{a^{2}}{b^{2}}}$$

Since \frac{b^{2}}{b^{2}} and \frac{a^{2}}{b^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{b^{2}+a^{2}}{b^{2}}}{\frac{b^{2}-a^{2}}{b^{2}}}$$

Divide \frac{b^{2}+a^{2}}{b^{2}} by \frac{b^{2}-a^{2}}{b^{2}} by multiplying \frac{b^{2}+a^{2}}{b^{2}} by the reciprocal of \frac{b^{2}-a^{2}}{b^{2}}.

$$\frac{\left(b^{2}+a^{2}\right)b^{2}}{b^{2}\left(b^{2}-a^{2}\right)}$$

Cancel out b^{2} in both numerator and denominator.

$$\frac{a^{2}+b^{2}}{-a^{2}+b^{2}}$$